Consider the Hilbert space
$$
\mathcal{H}:= L^2(S^1)
$$
with $S^1$ the unit circle.
On $\mathcal{H}$ let us introduce the equivalence relation
$$
f\sim g : \Leftrightarrow f(\cdot ) = g(\cdot + \alpha)\quad
\mbox{for some }\alpha \in S^1.
$$
Now define the factor space
$$
\overline{\mathcal{H}}:= \mathcal{H}/\sim.
$$
What is the structure of $\overline{\mathcal{H}}$? Is it a Hilbert manifold? If so, how to construct the smooth structure?
I am particularly interested in computing a (Riemannian) distance between two elements of $\overline{\mathcal{H}}$.

Not a Hilbert manifold. As the answer(s) have pointed out, you get problems at fixed points of the circle action (indeed, at any point where the circle action has a non-trivial stabiliser). There's another problem which is that the circle does not act continuously on the Hilbert space - how much of a problem this is will depend on how you want to fix the first problem (for example, you could go for a stratified space).
–
Loop SpaceJun 25 '12 at 9:28

1 Answer
1

It probably is NOT a smooth manifold. I think finding a chart around the point corresponding to constants, namely, the fixed points of the action of the group of rotation, is problematic.More precisely, at a fixed point, there is not a well-defined tangent space.